[PPT] - Foundations of Mathematics and Grundlagenkrise Vincent Steffan PowerPoint Presentation

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Foundations of Mathematics and Grundlagenkrise

Vincent Steffan 05.06.2018

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Introduction

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Euklid

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Euklid

Lived in the third century B.C.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Euklid

Lived in the third century B.C.
He tried to summarize all the mathematics done so far in

ancient Greece.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Euklid

Lived in the third century B.C.
He tried to summarize all the mathematics done so far in

ancient Greece.

For this he used a collection of postulates and axioms.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Postulates

”Let the following be postulated:

To draw a straight line from any point to any point.
To extend a finite straight line continuously in a straight line.
To describe a circle with any center and distance, the radius.
That all right angles are equal to one another.
(The so-called parallel postulate)That, if a straight line falling
n two straight lines make the interior angles on the same side

less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.”

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Axioms

Things that are equal to the same thing are also equal to one

another.

If equals are added to equals, then the wholes are equal.
If equals are subtracted from equals, then the remainders are

equal.

Things that coincide with one another are equal to one

another.

The whole is greater than the part.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

A first example of a ”Grundlagenkrise”

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

A first example of a ”Grundlagenkrise”

The Pythagoreans discovered, that there are numbers that are

not a fraction of two integers.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

A first example of a ”Grundlagenkrise”

The Pythagoreans discovered, that there are numbers that are

not a fraction of two integers.

An example: If you take a square with side length 1, the

diagonal has length √ 2, which is irrational.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Grundlagenkrise – The foundational crisis

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Georg Cantor

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Georg Cantor

Cantor wanted to define an axiomatic foundation of all

Mathematics.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Georg Cantor

Cantor wanted to define an axiomatic foundation of all

Mathematics.

He did this by establishing set theory in an axiomatic way.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Cantors naive set theory – the axioms

A set is any collection of definite, distinguishable objects of
ur intuition or of our intellect to be conceived as a whole

(i.e., regarded as a single unity).

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Cantors naive set theory – the axioms

A set is any collection of definite, distinguishable objects of
ur intuition or of our intellect to be conceived as a whole

(i.e., regarded as a single unity).

A set is completely determined by its members.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Cantors naive set theory – the axioms

A set is any collection of definite, distinguishable objects of
ur intuition or of our intellect to be conceived as a whole

(i.e., regarded as a single unity).

A set is completely determined by its members.
Every property determines a set.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Cantors naive set theory – the axioms

A set is any collection of definite, distinguishable objects of
ur intuition or of our intellect to be conceived as a whole

(i.e., regarded as a single unity).

A set is completely determined by its members.
Every property determines a set.
Given any set F of nonempty pairwise disjoint sets, there is a

set that contains exactly one member of each set in F.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Three paradoxa

Although this way of defining the term set is quite natural,

there occur paradoxa.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Three paradoxa

Although this way of defining the term set is quite natural,

there occur paradoxa.

Around the year 1900, Cesare Burali-Forti, Greogor Cantor

himself and Bertrand Russell found three paradoxa.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Cantors paradox: Cardinal numbers

Let A = {1, ..., n}. We say A has cardinal number n. Two sets A and B have the same cardinality, if there is a bijective function f : A → B. We say that the cardinality of A is greater or equal than the cardinality of B if there is a surjective function f : A → B. The cardinality of A is greater if it is greater of equal, but not equal to the cardinality of B.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Cantors paradox: Cardinal numbers

Theorem. Let A be a set and denote by 2A its powerset, i.e. the

set of all subsets of A. Then the cardinality of 2A is greater than the cardinality of A.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Cantors paradox

Let A be the set of all sets. Then – since all subsets of A are sets – we have 2A ⊂ A, which implies, that the cardinality of A is greater

r equal than the cardinality of 2A, which contradicts the theorem.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Schools of recovery

The mathematicians were shocked by the failure of naive set

theory.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Schools of recovery

The mathematicians were shocked by the failure of naive set

theory.

In order to get to a different foundation of mathematics, three

main schools were developed by the leading mathematicians

f this time.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Intuitionism

Main representatives of Intuitionism: The Dutch

mathematician Luitzen Brouwer and his student Arend Heyting.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Intuitionism

Main representatives of Intuitionism: The Dutch

mathematician Luitzen Brouwer and his student Arend Heyting.

Intuitionism sees mathematical objects as a part of the

intuition, as a part of the mind.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Intuitionism

Main representatives of Intuitionism: The Dutch

mathematician Luitzen Brouwer and his student Arend Heyting.

Intuitionism sees mathematical objects as a part of the

intuition, as a part of the mind.

They thought of infinite sets as potentially infinite. In an

infinite set we thus can find any number of elements, but we cannot work with all of them as infinitely many.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Intuitionism

Main representatives of Intuitionism: The Dutch

mathematician Luitzen Brouwer and his student Arend Heyting.

Intuitionism sees mathematical objects as a part of the

intuition, as a part of the mind.

They thought of infinite sets as potentially infifinite. In an

infinite set we thus can find any number of elements, but we cannot work with all of them as infinitely many.

The intuitionists did not believe in the principle of ”tertium

non datur”.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Logicism

Main representatives: Frege, Boole, Peano, Russell, and

Whitehead.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Logicism

Main representatives: Frege, Boole, Peano, Russell, and

Whitehead.

The logicistic approach on a foundation of mathematics was

described in the book ”Principia Mathematica” by Whitehead and Russell.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Logicism

Main representatives: Frege, Boole, Peano, Russell, and

Whitehead.

The logicistic approach on a foundation of mathematics was

described in the book ”Principia Mathematica” by Whitehead

The logicists tried to build mathematics on pure logic.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Logicism

Main representatives: Frege, Boole, Peano, Russell, and

Whitehead.

The logicistic approach on a foundation of mathematics was

described in the book ”Principia Mathematica” by Whitehead and Russell.

The logicists tried to build mathematics on pure logic.
They avoided all known paradoxa, but were not able to proof

consistency or completeness of this theory.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Logicism

Main representatives: Frege, Boole, Peano, Russell, and

Whitehead.

The logicistic approach on a foundation of mathematics was

described in the book ”Principia Mathematica” by Whitehead and Russell.

The logicists tried to build mathematics on pure logic.
They avoided all known paradoxa, but were not able to prove

consistency or completeness of this theory.

In this process some powerful tools were developed: As an

example, Peano was the first one to use symbols like ”∈” or ”⇒”.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Formalism

Main initiator of this theory: David Hilbert.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Formalism

Main initiator of this theory: David Hilbert.
For the formalists, every mathematical statement was a finite

sequence of symbols or language and is completely detached from the real world.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Formalism

Main initiator of this theory: David Hilbert.
For the formalists, every mathematical statement was a finite

sequence of symbols or language and is completely detached from the real world.

The formalists worked with so-called formal systems: A formal

system consists of:

A finite set of symbols to construct formulas
A decision procedure to decide whether a formula is true or

not.

A set of formulas assumed to be true, so-called axioms.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

G¨

dels incompleteness theorem

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

G¨

dels incompleteness theorem

In the 1930’s, Kurt G¨

del showed, that the goal to find a complete

and consistent foundation of mathematics cannot be reached.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Formal systems

A formal system again consists of a language with specified

well-defined statements, some axioms and some inference rules.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Formal systems

A formal system again consists of a language with specified

well-defined statements, some axioms and some inference rules.

A formal system T is called consistent, if there is no statement

A, such that both A and its negation follows from T .

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Formal systems

A formal system again consists of a language with specified

well-defined statements, some axioms and some inference rules.

A formal system T is called consistent, if there is no statement

A, such that both A and its negation follows from T .

A system is called complete, if for every well-defined

statement A either A or its negation follows from T .

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Formal systems

A formal system again consists of a language with specified

well-defined statements, some axioms and some inference rules.

A formal system T is called consistent, if there is no statement

A, such that both A and its negation follows from T .

A system is called complete, if for every well-defined

statement A either A or its negation follows from T .

A third property of a formal system is to be ”sufficiently

powerful”, which essentially means, that it can describe basic mathematical concepts as the natural numbers.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

Formal systems

A formal system again consists of a language with specified

well-defined statements, some axioms and some inference rules.

A formal system T is called consistent, if there is no statement

A, such that both A and its negation follows from T .

A system is called complete, if for every well-defined

statement A either A or its negation follows from T .

A third property of a formal system is to be ”sufficiently

powerful”, which essentially means, that it can describe basic mathematical concepts as the natural numbers.

The property ”recursive enumerability” prohibits things like

infinitely long proofs. Essentially this property means that every proof can be verified in a mechanical way (e.g. by a computer).

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

The incompleteness theorems

Theorem. Any sufficiently powerful, recursively enumerable

formal system is either inconsistent or incomplete.

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Introduction Grundlagenkrise – The foundational crisis Schools of recovery G¨

dels incompleteness theorem

The incompleteness theorems

Theorem. Any sufficiently powerful, recursively enumerable

formal system is either inconsistent or incomplete.

Theorem. Any sufficiently powerful consistent formal system